We present an open source computational framework geared towards the efficient numerical investigation of open quantum systems written in the Julia programming language. Built exclusively in Julia and based on standard quantum optics notation, the toolbox offers speed comparable to low-level statically typed languages, without compromising on the accessibility and code readability found in dynamic languages. After introducing the framework, we highlight its features and showcase implementations of generic quantum models. Finally, we compare its usability and performance to two well-established and widely used numerical quantum libraries. Nature of problem: Dynamics of open quantum systemsSolution method: Numerically solving the Schrödinger or master equation or a Monte Carlo wave-function approach. Additional comments including Restrictions and Unusual features:The framework may be used for problems that fulfill the necessary conditions such that they can be described by a Schrödinger or master equation. Furthermore, the aim is to efficiently and easily simulate systems of moderate size rather than pushing the limits of what is possible numerically.

Inherent binary or collective interactions in ensembles of quantum emitters induce a spread in the energy and lifetime of their eigenstates. While this typically causes fast decay and dephasing, in many cases certain special entangled collective states with minimal decay can be found, which possess ideal properties for spectroscopy, precision measurements or information storage. We show that for a specific choice of laser frequency, power and geometry or a suitable configuration of control fields one can efficiently prepare these states. We demonstrate this by studying preparation schemes for strongly subradiant entangled states of a chain of dipole-dipole coupled emitters. The prepared state fidelity and its entanglement depth is further improved via spatial excitation phase engineering or tailored magnetic fields.

A ring of sub-wavelength spaced dipole-coupled quantum emitters possesses only few radiant but many extraordinarily subradiant collective modes. These exhibit a 3D-confined spatial radiation field pattern forming a nano-scale high-Q optical resonator. We show that tailoring the geometry, orientation and distance between two such rings allows for increasing the ratio of coherent ring-toring coupling versus free-space emission by several orders of magnitude. In particular we find that subradiant excitations, when delocalized over several ring sites, are effectively transported between the rings with a high fidelity.

Ramsey interferometry is routinely used in quantum metrology for the most sensitive measurements of optical clock frequencies. Spontaneous decay to the electromagnetic vacuum ultimately limits the interrogation time and thus sets a lower bound to the optimal frequency sensitivity. In dense ensembles of two-level systems the presence of collective effects such as superradiance and dipole-dipole interaction tends to decrease the sensitivity even further. We show that by a redesign of the Ramsey-pulse sequence to include different rotations of individual spins that effectively fold the collective state onto a state close to the center of the Bloch sphere, partial protection from collective decoherence and dephasing is possible. This allows a significant improvement in the sensitivity limit of a clock transition detection scheme over the conventional Ramsey method for interacting systems and even for non-interacting decaying atoms.

Spontaneous emission of a two-level atom in free space is modified by other atoms in its vicinity leading to super-and subradiance. In particular, for atomic distances closer than the transition wavelength the maximally entangled antisymmetric superposition state of two individually excited atomic dipole moments possesses no total dipole moment and will not decay spontaneously at all. Such a two-atom dark state does not exist, if the atoms feature additional decay channels towards other lower energy states. However, we show here that for any atomic state with N − 1 independent spontaneous decay channels one can always find a N -particle highly entangled state, which completely decouples from the free radiation field and does not decay. Moreover, we show that this state is the unique state orthogonal to the subspace spanned by the lower energy states with this property. Its subradiant behavior largely survives also at finite atomic distances. The decay of an excited atomic state towards lower lying states via spontaneous emission is one of the most striking consequences of the quantum nature of the free radiation field [1]. Heuristically introduced even before e.g. by Einstein, the spontaneous emission rate Γ = Interestingly, it turns out that for several particles the emission process is not independent but can be collectively enhanced or reduced depending on the atomic arrangement [3]. It was already noted some time ago that these superradiant and subradiant collective states, where a single excitation is distributed over many particles, are entangled atomic states [4,5]. Although a recent classical coupled dipole model also leads to subradiancelike phenomena [6], the most superradiant and the perfect dark states for two two-level atoms with states (|g , |e ) are the maximally entangled symmetric and antisymmetric dipole moment superposition statesWhile superradiance on a chosen transition persists in the case, when the atom possesses more than one decay channel, there is no completely dark state for two multilevel atoms with several decay channels from a single excited state |e to several lower lying states |g i as schematically depicted in Fig. 1. Hence, in practise the observation of subradiant states is much more difficult than seeing superradiance, as all other decay channels need to be excluded [7].In this paper, we introduce a new class of subradiant or dark states appearing for atoms with several independent transitions. As a key result of this work we find that for systems of N particles one can construct highly * Laurin.Ostermann@uibk.ac.at multi-partite entangled states, where all N − 1 independent decay channels are suppressed. For these states the total dipole moments on all of these N − 1 transitions simultaneously vanish and at least in principle the optical excitation in this state will be stored indefinitely.After introducing our model atom system and the generalized, unique multi-atom dark states, we will discuss their special entanglement properties and possible quantum information processing...

Atoms trapped in magic wavelength optical lattices provide a Doppler-and collision-free dense ensemble of quantum emitters ideal for fast high precision spectroscopy and thus they are the basis of the best optical clock setups to date. Despite the minute optical dipole moments the inherent long range dipole-dipole interactions in such lattices at high densities generate measurable line shifts, increased dephasing and modified decay rates. We show that these effects can be resonantly enhanced or suppressed depending on lattice constant, geometry and excitation procedure. While these interactions generally limit the accuracy and precision of Ramsey spectroscopy, under optimal conditions collective effects can be exploited to yield zero effective shifts and long dipole lifetimes for a measurement precision beyond a noninteracting ensemble. In particular, 2D lattices with a lattice constant below the optical wavelength feature an almost ideal performance.Since the turn of the century the technology of manipulating and controlling ultracold atoms and molecules with laser light has seen breathtaking advances [1][2][3]. Following the seminal first demonstration of a quantum phase transition in an optical lattice [4], nowadays the so-called Mott insulator phase for atoms in an optical lattice can be prepared almost routinely [5, 6] and experiments with photo-associated ultracold molecules have reached a comparable level of control [7][8][9][10]. Employing the particles' internal structure coherent interactions between the atoms at neighboring sites in such a lattice can be tailored to a large extend, e.g. via spin-dependent tunneling [11].In one of the typical experimental setups atoms possessing a long-lived clock transition are prepared in an optical lattice using a differential light shift free (magic) trapping wavelength [12, 13]. Its most prominent application is the implementation of the world's best optical clocks [14][15][16].When the atoms in such a lattice are excited on an optical or infrared transition they will interact stronger and on a much longer range via dipole-dipole energy exchange than via tunneling or collisions. At sufficient densities the dipole interaction strength surpasses the excited state lifetime and resonantly enhanced collective excitations analogous to excitons in solid state physics appear [17, 18]. For polar molecules in optical lattices, such long wavelength collective excitations even dominate the dynamics [19] and can form the basis for studying generic phenomena of solid state physics by means of tailored toy models [1]. In the case of clock transitions, the extremely tiny dipole moment keeps these interactions small in absolute magnitude. Nevertheless, for densities close to unit filling the exciton's effective transition frequencies and their spontaneous decay is governed by dipole-dipole interaction [20] and deviates from the bare atom case.In a standard Ramsey interrogation sequence used as the generic basis for a clock setup, the first pulse creates a product state of all s...

An ideal superradiant laser on an optical clock transition of noninteracting cold atoms is predicted to exhibit an extreme frequency stability and accuracy far below mHz-linewidth. In any concrete setup sufficiently many atoms have to be confined and pumped within a finite cavity mode volume. Using a magic wavelength lattice minimizes light shifts and allows for almost uniform coupling to the cavity mode. Nevertheless, the atoms are subject to dipole-dipole interaction and collective spontaneous decay which compromises the ultimate frequency stability. In the high density limit the Dicke superradiant linewidth enhancement will broaden the laser line and nearest neighbor couplings will induce shifts and fluctuations of the laser frequency. We estimate the magnitude and scaling of these effects by direct numerical simulations of few atom systems for different geometries and densities. For Strontium in a regularly filled magic wavelength configuration atomic interactions induce small laser frequency shifts only and collective spontaneous emission weakly broadens the laser. These interactions generally enhance the laser sensitivity to cavity length fluctuations but for optimally chosen operating conditions can lead to an improved synchronization of the atomic dipoles.

Energy and lifetime of collective optical excitations in regular arrays of atoms and molecules are significantly influenced by dipole-dipole interaction. While the dynamics of closely positioned atoms can be approximated well by the Dicke superradiance model, the situation of finite regular configurations is hard to access analytically. Most treatments use an exciton based description limited to the lowest excitation manifold. We present a general approach studying the complete decay cascade of a finite regular array of atoms from the fully inverted to the ground state. We explicitly calculate all energy shifts and decay rates for two generic cases of a three-atom linear chain and an equilateral triangle. In numerical calculations we show that despite fairly weak dipole-dipole interactions, collective vacuum coupling allows for superradiant emission as well as subradiant states in larger arrays through multi-particle interference. This induces extra dephasing and modified decay as important limitations for Ramsey experiments in lattice atomic clock setups as well as for the gain and frequency stability of superradiant lasers.